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Line 1: − + − The foundations of real analysis were shaken at the end of the 19th century with the work of [[Richard Dedekind]], who relaid the Archimedean-style groundwork with his own radical concepts. His [[Dedekind cut]]s undercut the assumption of the continuity of the real line, by ''cutting'' at ''gaps'' between points. Mathematicians were worried that his techniques used the dubious [[Axiom of Choice]] and seemingly [[elementary_proof|non-elementary]] methods. However, the unifying ideas of [[Cauchy]], specifically that of the [[Cauchy sequence]] and [[complete metric space|completeness]], helped eliminate doubts and gain acceptance for Dedekind's ideas among real analysts. Dedekind cuts are now viewed as a solid foundation for real analysis, more than Archimedes' ideas ever were.<ref>http://plato.stanford.edu/entries/dedekind-foundations/#FouAna</ref>+ − ====References====+ − + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Start of rewrite. Took out mistaken paragraph, it has already been discussed in talk.
'''Real analysis''' is a field in [[mathematics]] that focuses on the set of [[real number]]s, their properties, [[sequence]]s and [[function]]s. Included in this branch of mathematics is concepts of [[Limit_%28mathematics%29|limit]]s and [[convergence]], [[calculus]], and properties of real-valued functions such as [[continuous|continuity]].
'''Real analysis''' is a field in [[mathematics]] that focuses on the set of [[real number]]s, their properties, [[sequence]]s and [[function]]s. Included in this branch of mathematics is concepts of [[Limit_%28mathematics%29|limit]]s and [[convergence]], [[calculus]], and properties of real-valued functions such as [[continuous|continuity]]. It also includes [[measure theory]].
For the purposes of this article, "analysis" will be limited to the generalization and extension of the concepts of calculus, using the concepts of elementary [[point-set topology]].
The reader should be quite familiar with the concepts of calculus, especially limits and continuity. In particular, the reader should be comfortable with the ramifications of the phrase "for every epsilon".
{{reflist}}
==Open sets==
Open sets (and, by extension, closed sets, which are just the complements of open sets) are the fundamental concept of analysis. Analysis and topology are really just the study of open sets.
Before giving the definition of open sets in Euclidean space, we present some examples. Readers who are aware of the general inuitive notion of open sets should find these examples familiar.
The simplest open sets in 1-dimensional Euclidean space (formally called <math>\mathbb{R}^1</math>; informally called the real numbers of the "real line") are '''open intervals'''. An open interval consists of those numbers lying strictly between two endpoints a and b. In set-theoretic notation:
::<math>\{ x\ |\ a < x < b \}\,</math>
A shorter notation for this set consists of the two endpoints in parentheses:
::<math>( a, b )\,</math>
A '''closed interval''' (we will have more to say about closed sets later) would include the endpoints. It is commonly denoted with brackets:
::<math>[ a, b ] = \{ x\ |\ a \le x \le b \}</math>
An interval that includes one endpoint but not the other is called semi-open:
::<math>[ a, b ) = \{ x\ |\ a \le x < b \}</math>
::<math>( c, d ] = \{ x\ |\ c < x \le d \}</math>
When drawing pictures of intervals, those same symbols are typically used:
::::''Need a picture here!''
Open intervals are not the only open sets. Any union of open intervals is an open set. For example:
::<math>\{ N, N+1/N\ |\ N \textrm{\ is\ an\ integer} \ge 2 \}</math>
Bizarrely defined sets like the one above are commonly used as examples and counterexamples in analysis and topology.
[[category:mathematics]]
[[category:mathematics]]